class: center, middle, inverse, title-slide .title[ # Statistical Concepts Everyone Should Know ] .subtitle[ ## <p><em>Statistics for Life</em></p> ] .author[ ### <p><strong>John Slough</strong><br /> <em>for</em> <strong>The John & Calvin Podcast</strong></p> ] --- class: inverse, center, large <h1 style="font-size: 80px; margin-top: 150px;">Beware the Metrics</h1> <hr style="margin-top: 2em; margin-bottom: 2em;"> -- <h2 style="font-size: 50px;">Misuse and Misunderstanding, Not Math, Mislead</h2> -- <h2 style="font-size: 40px;">mean ≠ median</h2> --- ## 1. Mean vs Median ### Local bar weath distribution: 50 Patrons <div id="htmlwidget-983315589fed18d622a7" style="width:750px;height:450px;" class="plotly html-widget"></div> <script type="application/json" 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style='color:#C04E3D'><b>Mean:<\/b><\/span> $60,335<br><span style='color:#BFA06D'><b>Median:<\/b><\/span> $59,441","showarrow":false,"align":"right","font":{"size":18,"color":"black"}}]},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"105622e29496":{"x":{},"type":"bar"},"1056498915be":{"xintercept":{}},"105646f292a9":{"xintercept":{}}},"cur_data":"105622e29496","visdat":{"105622e29496":["function (y) ","x"],"1056498915be":["function (y) ","x"],"105646f292a9":["function (y) 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Mean vs Median ### Local bar weath distribution: 50 Patrons + Billon Gezos <div id="htmlwidget-6f1d8b9e786d9d3fe28d" style="width:750px;height:450px;" class="plotly html-widget"></div> <script type="application/json" 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style='color:#C04E3D'><b>Mean:<\/b><\/span> $2,941,235,622<br><span style='color:#BFA06D'><b>Median:<\/b><\/span> $59,505","showarrow":false,"align":"right","font":{"size":18,"color":"black"}}]},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"1056a2ab54f":{"x":{},"type":"bar"},"105679e55abf":{"xintercept":{}}},"cur_data":"1056a2ab54f","visdat":{"1056a2ab54f":["function (y) ","x"],"105679e55abf":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ## 1. Mean vs Median <div style="font-size: 130%; text-align: left; margin: 100px 0;"> <p><strong>Mean</strong>: Sum of all values divided by the number of values.</p> <br> <p><strong>Median</strong>: Middle value when the values are ordered.</p> </div> --- ## 1. Mean vs Median ### Impact of a $150B Outlier on Mean vs. Median — Across Sample Sizes <table class="table" style="font-size: 24px; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:right;"> Sample Size </th> <th style="text-align:right;"> Mean </th> <th style="text-align:right;"> Median </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 101 </td> <td style="text-align:right;padding-left: 20px;"> $1,485,214,194 </td> <td style="text-align:right;padding-left: 20px;"> $61,587 </td> </tr> <tr> <td style="text-align:right;"> 1,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 149,915,313 </td> <td style="text-align:right;padding-left: 20px;"> $60,141 </td> </tr> <tr> <td style="text-align:right;"> 10,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 15,063,285 </td> <td style="text-align:right;padding-left: 20px;"> $59,610 </td> </tr> <tr> <td style="text-align:right;"> 100,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 1,564,864 </td> <td style="text-align:right;padding-left: 20px;"> $59,897 </td> </tr> <tr> <td style="text-align:right;"> 1,000,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 214,846 </td> <td style="text-align:right;padding-left: 20px;"> $59,868 </td> </tr> <tr> <td style="text-align:right;"> 10,000,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 79,864 </td> <td style="text-align:right;padding-left: 20px;"> $59,885 </td> </tr> <tr> <td style="text-align:right;"> 100,000,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 66,365 </td> <td style="text-align:right;padding-left: 20px;"> $59,876 </td> </tr> </tbody> </table> --- ## 1. Mean vs Median A more realistic example  --- .pull-left[ ### Economics & Wealth - GDP per capita - household income - net worth - home price - monthly rent - CEO compensation ### Health & Healthcare - life expectancy - healthcare spending per person - hospital bill - patient out-of-pocket cost ] .pull-right[ ### Academics - test score (SAT, PISA) - GPA - academic citations - speaking invitations per expert ### Other - commute time - screen-time per user - household energy use - carbon emissions per capita - YouTube ad revenue per channel - revenue per app in the App Store - Software Bug Fix Times ] --- ## 1. Mean vs Median ### Pareto Distribution <div id="htmlwidget-dcdfc64296f5794e54a4" style="width:750px;height:450px;" class="plotly html-widget"></div> <script type="application/json" 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Mean vs Median ### Normal Distribution <div id="htmlwidget-cbd3ad10bd2d35a2e76d" style="width:750px;height:450px;" class="plotly html-widget"></div> <script type="application/json" 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style='color:#C04E3D'><b>Mean:<\/b><\/span> $90,086<br><span style='color:#BFA06D'><b>Median:<\/b><\/span> $90,024","showarrow":false,"align":"right","font":{"size":18,"color":"black"}}]},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"10562ea66fb9":{"x":{},"type":"bar"},"105630f0f29c":{"xintercept":{}},"10561ac7f8de":{"xintercept":{}}},"cur_data":"10562ea66fb9","visdat":{"10562ea66fb9":["function (y) ","x"],"105630f0f29c":["function (y) ","x"],"10561ac7f8de":["function (y) 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Mean vs Median <br> ## Extreme Outliers <br> -- ## Symmetric vs Skewed <br> -- ## the Shape of the Data --- class: inverse, center, large <h1 style="font-size: 80px; margin-top: 200px;">Distributions</h1> <hr style="margin-top: 2em; margin-bottom: 2em;"> -- <h2 style="font-size: 60px;">The Foundation Beneath Every Statistic</h2> --- ## 2. Distributions ### Theoretical models of where values are likely to occur - **Symmetry vs. Skewness** Are values evenly spread around the center, or do they stretch more in one direction? - **Variability (Spread)** How tightly or widely are the values clustered? - **Tail Behavior** How likely are extreme or outlier values? ---  ---  --- ## 2. Distributions $$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \, e^{ -\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2 } $$ Where: - `\(\mu\)` = the **mean** (center) - `\(\sigma\)` = the **standard deviation** (spread) - `\(x\)` = the value at which we evaluate the density - `\(f(x)\)` = the probability density at `\(x\)` given `\(\mu\)` and `\(\sigma\)` --- .center[ **Normal distribution** ] $$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \, e^{ -\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2 } $$ <hr style="border: 0; height: 1px; background: lightgray;"> <div style="text-align: center;"> <img src="assets/normal_annotated.png"> </div> --- ## 2. Distributions - Outcomes fall across possibilities - Those possibilities form a distribution - Identifying the distribution can shape your analysis --- class: inverse, center, large <h1 style="font-size: 80px; margin-top: 150px;">Correlation, Confounding, Causation</h1> <hr style="margin-top: 2em; margin-bottom: 2em;"> -- <h2 style="font-size: 80px;">and the Stories We Tell</h2> --- class: inverse, center, large, middle <h1 style="font-size: 80px; ">Correlation</h1> ---  ---  --- class: inverse, center, large, middle <h1 style="font-size: 80px; ">Confounding</h1> --- .pull-left[ <div class="plotly html-widget html-fill-item" id="htmlwidget-f25d4e1770d66a86dcce" style="width:360px;height:288px;"></div> <script type="application/json" 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(g/day)","yaxis":"y","line":{"color":"black","width":5},"marker":{"color":"rgba(255,127,14,1)","line":{"color":"rgba(255,127,14,1)"}},"error_y":{"color":"rgba(255,127,14,1)"},"error_x":{"color":"rgba(255,127,14,1)"},"xaxis":"x","frame":null},{"x":[1901,1902,1903,1904,1905,1906,1907,1908,1909,1910,1911,1912,1913,1914,1915,1916,1917,1918,1919,1920,1921,1922,1923,1924,1925,1926,1927,1928,1929,1930,1931,1932,1933,1934,1935,1936,1937,1938,1939,1940,1941,1942,1943,1944,1945,1946,1947,1948,1949,1950,1951,1952,1953,1954,1955,1956,1957,1958,1959,1960,1961,1962,1963,1964,1965,1966,1967,1968,1969,1970,1971,1972,1973,1974,1975,1976,1977,1978,1979,1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020],"y":[20,21,25,30,40,40,50,55,57,60,63,65,70,80,90,90,100,110,115,120,120,130,145,150,160,165,175,188,200,220,225,230,240,250,255,270,290,310,325,345,355,360,375,385,390,400,410,420,430,440,460,480,505,540,550,570,585,590,595,600,620,630,640,650,660,670,685,695,705,715,720,730,740,735,740,730,720,715,710,715,720,725,735,740,750,760,770,760,765,765,760,750,735,725,710,715,720,730,725,725,720,720,710,710,705,700,685,670,655,640,630,615,610,600,600,605,605,605,610,610],"type":"scatter","mode":"lines","name":"Heart Disease Deaths (thousands)","yaxis":"y2","line":{"color":"red","width":5},"marker":{"color":"rgba(44,160,44,1)","line":{"color":"rgba(44,160,44,1)"}},"error_y":{"color":"rgba(44,160,44,1)"},"error_x":{"color":"rgba(44,160,44,1)"},"xaxis":"x","frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> <small style="font-size:12px; line-height:1.1; display:inline-block;"> "Figure 2 correlates saturated fat and total vegetable oil consumption versus heart disease deaths in the U.S.A., with data on all three dating back to at least 1909."</small> <small style="font-size:12px; line-height:1; display:inline-block;"> Data: <a href="https://www.fns.usda.gov/cnpp/us-foodsupply/nutrient-content-1909-2010">USDA Food Supply</a> | Paper: <a href="https://www.sciencedirect.com/science/article/abs/pii/S0306987717305017?via%3Dihub">ScienceDirect</a> | Video: <a href="https://www.youtube.com/watch?v=7kGnfXXIKZM">YouTube</a> </small> --- class: inverse, center, large <h1 style="font-size: 80px; margin-top: 200px;">Shapes of Change</h1> <hr style="margin-top: 2em; margin-bottom: 2em;"> -- <h2 style="font-size: 60px;">What Curve, What Point?</h2> --- ## 4. Shapes of Change <div id="htmlwidget-4cadc5f8c7ab9ff923ac" style="width:750px;height:500px;" class="plotly html-widget"></div> <script type="application/json" 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Bias **Bias** is the difference between the expected value of an estimator and the true value of the parameter it estimates. Formally: $$ \text{Bias}(\hat{\theta}) = \mathbb{E}[\hat{\theta}] - \theta $$ Where: - `\(\hat{\theta}\)` = the **estimator** (your calculated estimate) - `\(\theta\)` = the **true parameter** (the real value you want) - `\(\mathbb{E}[\hat{\theta}]\)` = the **expected value** of the estimator (its long-run average over many samples) - If Bias = 0 → the estimator is **unbiased**. - If Bias ≠ 0 → the estimator is **biased** (systematically too high or too low). --- Bias doesn’t just shift your first guess. It filters what you see next — making you even more biased over time Selection bias: The sample is not representative of the population. Omitted variable bias: Leaving out a variable that influences both the dependent and independent variables. Measurement bias (or information bias): Errors in how data is collected or recorded. Survivorship bias: Only analyzing "survivors" or those who remain, ignoring those who dropped out or failed. Recall bias: Errors because people remember things inaccurately (common in surveys and retrospective studies). Observer bias: Researcher's expectations subtly influence measurements or observations. Publication bias: Studies with "positive" results are more likely to be published than "null" or "negative" results. --- ## 5. Bias The Self-Reinforcing Feedback Loop of Bias <div style="display:flex; justify-content:center;"> <div id="htmlwidget-fc8df40fd67c2027a709" style="width:525px;height:525px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-fc8df40fd67c2027a709">{"x":{"diagram":"\n digraph bias_loop {\n graph [layout = circo]\n node [shape = box, style = \"rounded,filled\", fontname = Helvetica, fontsize = 12]\n\n CurrentBias [label = \"Current Bias\", fillcolor = \"#F5C4BB\"]\n SelectiveAttention[label = \"Selective Attention\",fillcolor = \"#EDA89C\"]\n SelectiveEvidence [label = \"Selective Evidence\", fillcolor = \"#E17A66\"]\n ReinforcedBias [label = \"Reinforced Bias\", fillcolor = \"#C04E3D\"]\n\n edge [penwidth = 2]\n\n CurrentBias -> SelectiveAttention\n SelectiveAttention-> SelectiveEvidence\n SelectiveEvidence -> ReinforcedBias\n ReinforcedBias -> CurrentBias\n }\n ","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </div> --- ### Consequence of the Self-Reinforcing Feedback Loop of Bias Conceptual Edition <div id="htmlwidget-d8c7d17cf4080686b93c" style="width:700px;height:500px;" class="plotly html-widget"></div> <script type="application/json" data-for="htmlwidget-d8c7d17cf4080686b93c">{"x":{"visdat":{"10563bc47a8a":["function () ","plotlyVisDat"],"1056604918ad":["function () ","data"],"10565ef3373d":["function () 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